Lattice Strips Research Articles

Exact results for average cluster numbers in bond percolation on infinite-length lattice strips.

We calculate exact analytic expressions for the average cluster numbers 〈k〉_{Λ_{s}} on infinite-length strips Λ_{s}, with various widths, of several different lattices, as functions of the bond occupation probability p. It is proved that these expressions are rational functions of p. As special cases of our results, we obtain exact values of 〈k〉_{Λ_{s}} and derivatives of 〈k〉_{Λ_{s}} with respect to p, evaluated at the critical percolation probabilities p_{c,Λ} for the corresponding infinite two-dimensional lattices Λ. We compare these exact results with an analytic finite-size correction formula and find excellent agreement. We also analyze how unphysical poles in 〈k〉_{Λ_{s}} determine the radii of convergence of series expansions for small p and for p near to unity. Our calculations are performed for infinite-length strips of the square, triangular, and honeycomb lattices with several types of transverse boundary conditions.

A variable-topology morphing composite cylindrical lattice

Morphing composite structures are of significant interest due to the fact that they exhibit superior mass-to-stiffness ratios and a large degree of tailorability in comparison to traditional materials and structures. One such morphing composite structure is the multistable composite cylindrical lattice. Current work introduces a novel variable-topology morphing mechanism to it through the use of both permanent magnets and electromagnets. By replacing a set of mechanical fasteners from the central intersection of the lattice strips with a bespoke variable-topology mechanism introduces a controllable and replicable semi-autonomous means for topology morphing. The variable-topology mechanism allows the structure to transition from being a linear deployment actuator to one that deploys along a curved path, without need for external mechanical input. The behaviour of both the variable-topology mechanism and the topology-changing cylindrical lattice are thoroughly characterised through a combination of mechanical and virtual tests.

Spectral entropy and strain energy trends in composite mechanical metamaterials

The strain energy and spectral (Shannon) entropy distributions in composite lattices made of core and reinforcement components are investigated by conducting numerical experiments on a set of indefinitely tall lattice strips subjected to a point surface load. The spectral entropy measures the complexity of the strain energy spectrum as it transforms with distance from the loaded surface. The spectral entropy behavior is tailored by adjusting the degree of anisotropy in the lattices. Isotropic continuum-like responses exhibited enhanced spectral entropy decay near the loaded surface and were associated with dispersion of strain energy over a wide spatial area of the lattice. Highly anisotropic responses showed a slow entropy decay at the surface and the localization of strain energy. Interestingly, with sufficient distance from the loaded surface, all lattice designs exhibited the same asymptotic rate of entropy decay, and this rate also was similar to an isotropic continuum material behavior. This implies in practice that any exotic properties of metamaterials determined by their modal selectivity are generally better pronounced in the vicinity of loads, and that they tend to diminish on the periphery of these interesting material systems.

On linear waveguides of square and triangular lattice strips: an application of Chebyshev polynomials

An analysis of the linear waves in infinitely-long square and triangular lattice strips of identical particles with nearest neighbour interactions for all combinations of fixed and free boundary conditions, as well as the periodic boundary, is presented. Expressions for the dispersion relations and the associated normal modes in these waveguides are provided in the paper; some of which are expressed implicitly in terms of certain linear combinations of the Chebyshev polynomials. The effect of next-nearest-neighbour interaction is also included for the square lattice waveguides. It is found that localized propagating waves, so called surface wave modes, occur in the triangular lattice strips, as well as square lattice strips with next-nearest-neighbour interactions, when either or both boundaries are free. In this paper, the even and odd modes are also discussed separately, wherever applicable. Graphical illustrations of the dispersion curves are included for all waveguides. The discrete waveguides analysed in the paper have broad applications in physics and engineering, including their merit in classical problems in elasticity, acoustics and electromagnetism, as well as recent technological issues involving various transport phenomena in quasi-one-dimensional nano-structures.

On linear waveguides of zigzag honeycomb lattice

An analysis of the scalar linear waves in infinite honeycomb lattice strips with discrete Dirichlet and Neumann boundary conditions, as well as the periodic boundary condition, is presented for the zigzag orientation. The dispersion relations and the associated wave modes in these waveguides are provided in the paper; the former in terms of the Chebyshev polynomials. It is found that the localized propagating waves, addressed as surface wave modes, occur in case of certain boundary conditions only; for which the details are provided separately. The dispersion relations and wave modes for certain even and odd symmetry are also discussed in the paper, wherever applicable. Graphical illustrations of the dispersion curves for all waveguides are included. Applications include the transmission of phononic, electronic, magnetic, and photonic waves in nanostructures of carbon-like constituents. The analytical results are directly applicable for the out-of-plane phonons assuming only nearest-neighbor interactions in graphene nanoribbons on certain substrates, as well as the counterparts in the single orbital nearest neighbor tight-binding model for the electrons.

DETERMINATION OF THE CYTOTOXICITY OF NEW NANOCOMPOSITES BASED ON SAPONITE

Development of new nanomaterials is a relevant area of research. However, the specific physical and chemical properties of nanomaterials compared to conventional microparticles can carry unexpected risks to the ecosystem. Therefore, the newly created research of nanomaterials currently is a very urgent task. The article deals with the cytotoxicity of new Nb-containing nanocomposites and nanomaterials SiO2, which is included in their composition. HR-TEM micrographs show that all Nb (V) containing nanocomposites consist of relatively small, sheet-like particles at nanometer scale. Nanocomposites Nb-Saponite (Et) samples are composed of particles with size is below 100 nm, smaller than the Nb-Saponite (Cl) sample. This difference in a crystal size could be related to different procedure adopted to prepare the solids. Photomicrographs obtained at higher magnification showed basal plane lattice strips of nanocomposites, their interlayer space was about 1.3 nm. The study has found that new nanocomposites show no toxicity, unlike their part – nanosized materials SiO2. The results of studies on the cytotoxic effect on cells of A. cepa L. have shown that nanosized materials SiO2 leads to a decrease in mitotic index and chromosomal disorders structures, while not causing nanocomposites toxic effect. The research has also revealed that the action of nanosized materials SiO2 concentrations in the range of 450-600 mg/l changed the duration of mitosis phase, growth of prophase unit was observed while ana-phase and telophase units decreased. Thus, we should conclude that the cytotoxicity of nanosized materials due to their size, which cause significant chemical activity and high capacity for penetration of nanoparticles into plant organisms.

On energy balance and the structure of radiated waves in kinetics of crystalline defects

Traveling waves, with well-known closed form expressions, in the context of the defects kinetics in crystals are excavated further with respect to their inherent structure of oscillatory components. These are associated with, so called, Frenkel–Kontorova model with a piecewise quadratic substrate potential, corresponding to the symmetric as well as asymmetric energy wells of the substrate, displacive phase transitions in bistable chains, and brittle fracture in triangular lattice strips under mode III conditions. The paper demonstrates that the power expended theorem holds so that the sum of rate of working and the rate of total energy flux into a control strip moving steadily with the defect equals the rate of energy sinking into the defect, in the sense of N.F. Mott. In the conservative case of the Frenkel–Kontorova model with asymmetric energy wells, this leads to an alternative expression for the mobility in terms of the energy flux through radiated lattice waves. An application of the same to the case of martensitic phase boundary and a crack, propagating uniformly in bistable chains and triangular lattice strips, respectively, is also provided and the energy release is expressed in terms of the radiated energy flux directly. The equivalence between the well-known expressions and their alternative is established via an elementary identity, which is stated and proved in the paper as the zero lemma. An intimate connection between the three distinct types of defects is, thus, revealed in the framework of energy balance, via a structural similarity between the corresponding variants of the ‘zero’ lemma containing the information about radiated energy flux. An extension to the dissipative models, in the presence of linear viscous damping, is detailed and analog of the zero lemma is proved. The analysis is relevant to the dynamics of dislocations, brittle cracks, and martensitic phase boundaries, besides possible applications to analogous physical contexts which are marked by macroscopic energy release through emission of waves and possibly linear viscous damping.

Wave Propagation in Bifurcated Waveguides of Square Lattice Strips

The propagation of waves in square lattice strips, as waveguides, with discrete analogues of hard and soft outer boundaries is considered in the presence of a bifurcating lattice row which is either soft or hard. By an application of the discrete Fourier transform, the related boundary value problem, for both types of the bifurcated waveguides, is formulated as a discrete Wiener--Hopf equation. The exact solution of the latter is found in the form of a contour integral on the unit circle in complex plane. Closed form expressions for the approximate solution in the far-field are provided using the normal mode expansion. A general expression for the reflectance and transmittance is obtained; the sum, in the absence of dissipation, is established as unity. It is shown that the limiting wave field, as the waveguide width tends to infinity, coincides with that on an infinite lattice; the proof is presented in the dissipative case. The discrete paradigm of the bifurcated waveguides, introduced in the paper, has natural applications in engineering and science.

Parallel family trees for transfer matrices in the Potts model

The computational cost of transfer matrix methods for the Potts model is related to the question in how many ways can two layers of a lattice be connected? Answering the question leads to the generation of a combinatorial set of lattice configurations. This set defines the configuration space of the problem, and the smaller it is, the faster the transfer matrix can be computed. The configuration space of generic (q,v) transfer matrix methods for strips is in the order of the Catalan numbers, which grows asymptotically as O(4m) where m is the width of the strip. Other transfer matrix methods with a smaller configuration space indeed exist but they make assumptions on the temperature, number of spin states, or restrict the structure of the lattice. In this paper we propose a parallel algorithm that uses a sub-Catalan configuration space of O(3m) to build the generic (q,v) transfer matrix in a compressed form. The improvement is achieved by grouping the original set of Catalan configurations into a forest of family trees, in such a way that the solution to the problem is now computed by solving the root node of each family. As a result, the algorithm becomes exponentially faster than the Catalan approach while still highly parallel. The resulting matrix is stored in a compressed form using O(3m×4m) of space, making numerical evaluation and decompression to be faster than evaluating the matrix in its O(4m×4m) uncompressed form. Experimental results for different sizes of strip lattices show that the parallel family trees (PFT) strategy indeed runs exponentially faster than the Catalan Parallel Method (CPM), especially when dealing with dense transfer matrices. In terms of parallel performance, we report strong-scaling speedups of up to 5.7× when running on an 8-core shared memory machine and 28× for a 32-core cluster. The best balance of speedup and efficiency for the multi-core machine was achieved when using p=4 processors, while for the cluster scenario it was in the range p∈[8,10]. Because of the parallel capabilities of the algorithm, a large-scale execution of the parallel family trees strategy in a supercomputer could contribute to the study of wider strip lattices.

Topological Edge States and Fractional Quantum Hall Effect from Umklapp Scattering

We study anisotropic lattice strips in the presence of a magnetic field in the quantum Hall effect regime. At specific magnetic fields, causing resonant umklapp scattering, the system is gapped in the bulk and supports chiral edge states in close analogy to topological insulators. In electron gases with stripes, these gaps result in plateaus for the Hall conductivity exactly at the known fillings n/m (both positive integers and m odd) for the integer and fractional quantum Hall effect. For double strips, we find topological phase transitions with phases that support midgap edge states with flat dispersion. The topological effects predicted here could be tested directly in optical lattices.

Ferrimagnetism of the Heisenberg Models on the Quasi-One-Dimensional Kagome Strip Lattices

We study the ground-state properties of the S =1/2 Heisenberg models on the quasi-one-dimensional kagome strip lattices by the exact diagonalization and density matrix renormalization group methods. The models with two different strip widths share the same lattice structure in their inner part with the spatially anisotropic two-dimensional kagome lattice. When there is no magnetic frustration, the well-known Lieb–Mattis ferrimagnetic state is realized in both models. When the strength of magnetic frustration is increased, on the other hand, the Lieb–Mattis-type ferrimagnetism is collapsed. We find that there exists a non-Lieb–Mattis ferrimagnetic state between the Lieb–Mattis ferrimagnetic state and the nonmagnetic ground state. The local magnetization clearly shows an incommensurate modulation with long-distance periodicity in the non-Lieb–Mattis ferrimagnetic state. The intermediate non-Lieb–Mattis ferrimagnetic state occurs irrespective of strip width, which suggests that the intermediate phase of the two-dimensional kagome lattice is also the non-Lieb–Mattis-type ferrimagnetism.

Potts model on recursive lattices: some new exact results

We compute the partition function of the Potts model with arbitrary values of q and temperature on some strip lattices. We consider strips of width L y = 2, for three different lattices: square, diced and ‘shortest-path’ (to be defined in the text). We also get the exact solution for strips of the Kagome lattice for widths L y = 2,3,4,5. As further examples we consider two lattices with different type of regular symmetry: a strip with alternating layers of width L y = 3 and L y = m + 2, and a strip with variable width. Finally we make some remarks on the Fisher zeros for the Kagome lattice and their large q-limit.

Ground state entropy of the Potts antiferromagnet on homeomorphic expansions of kagomé lattice strips

We present exact calculations of the chromatic polynomial and resultant ground state entropy of the q-state Potts antiferromagnet on lattice strips that are homeomorphic expansions of a strip of the kagomé lattice. The dependence of the ground state entropy on the form of homeomorphic expansion is elucidated.

Bulk, surface and corner free-energy series for the chromatic polynomial on the square and triangular lattices

We present an efficient algorithm for computing the partition function of the q-colouring problem (chromatic polynomial) on regular two-dimensional lattice strips. Our construction involves writing the transfer matrix as a product of sparse matrices, each of dimension ∼3m, where m is the number of lattice spacings across the strip. As a specific application, we obtain the large-q series of the bulk, surface and corner free energies of the chromatic polynomial. This extends the existing series for the square lattice by 32 terms, to order q−79. On the triangular lattice, we verify Baxter's analytical expression for the bulk free energy (to order q−40), and we are able to conjecture exact product formulae for the surface and corner free energies.

Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips

We determine the general structure of the partition function of the $q$-state Potts model in an external magnetic field, $Z(G,q,v,w)$ for arbitrary $q$, temperature variable $v$, and magnetic field variable $w$, on cyclic, M\"obius, and free strip graphs $G$ of the square (sq), triangular (tri), and honeycomb (hc) lattices with width $L_y$ and arbitrarily great length $L_x$. For the cyclic case we prove that the partition function has the form $Z(\Lambda,L_y \times L_x,q,v,w)=\sum_{d=0}^{L_y} \tilde c^{(d)} Tr[(T_{Z,\Lambda,L_y,d})^m]$, where $\Lambda$ denotes the lattice type, $\tilde c^{(d)}$ are specified polynomials of degree $d$ in $q$, $T_{Z,\Lambda,L_y,d}$ is the corresponding transfer matrix, and $m=L_x$ ($L_x/2$) for $\Lambda=sq, tri (hc)$, respectively. An analogous formula is given for M\"obius strips, while only $T_{Z,\Lambda,L_y,d=0}$ appears for free strips. We exhibit a method for calculating $T_{Z,\Lambda,L_y,d}$ for arbitrary $L_y$ and give illustrative examples. Explicit results for arbitrary $L_y$ are presented for $T_{Z,\Lambda,L_y,d}$ with $d=L_y$ and $d=L_y-1$. We find very simple formulas for the determinant $det(T_{Z,\Lambda,L_y,d})$. We also give results for self-dual cyclic strips of the square lattice.

Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density

The classical monomer-dimer model in two-dimensional lattices has been shown to belong to the "#P-complete" class, which indicates the problem is computationally "intractable." We use exact computational method to investigate the number of ways to arrange dimers on mxn two-dimensional rectangular lattice strips with fixed dimer density rho . For any dimer density 0<rho<1 , we find a logarithmic correction term in the finite-size correction of the free energy per lattice site. The coefficient of the logarithmic correction term is exactly -12 . This logarithmic correction term is explained by the newly developed asymptotic theory of Pemantle and Wilson. The sequence of the free energy of lattice strips with cylinder boundary condition converges so fast that very accurate free energy f{2}(rho) for large lattices can be obtained. For example, for a half-filled lattice, f{2}(12)=0.633195588930 , while f{2}(14)=0.4413453753046 and f{2}(34)=0.64039026 . For rho<0.65 , f{2}(rho) is accurate at least to ten decimal digits. The function f{2}(rho) reaches the maximum value f{2}(rho{*})=0.662798972834 at rho{*}=0.6381231 , with 11 correct digits. This is also the monomer-dimer constant for two-dimensional rectangular lattices. The asymptotic expressions of free energy near close packing are investigated for finite and infinite lattice widths. For lattices with finite width, dependence on the parity of the lattice width is found. For infinite lattices, the data support the functional form obtained previously through series expansions.

Partition function zeros of a restricted Potts model on lattice strips and effects of boundary conditions

We calculate the partition function $Z(G,Q,v)$ of the $Q$-state Potts model exactly for strips of the square and triangular lattices of various widths $L_y$ and arbitrarily great lengths $L_x$, with a variety of boundary conditions, and with $Q$ and $v$ restricted to satisfy conditions corresponding to the ferromagnetic phase transition on the associated two-dimensional lattices. From these calculations, in the limit $L_x \to \infty$, we determine the continuous accumulation loci ${\cal B}$ of the partition function zeros in the $v$ and $Q$ planes. Strips of the honeycomb lattice are also considered. We discuss some general features of these loci.

Logarithmic corrections in the free energy of monomer-dimer model on plane lattices with free boundaries

Using exact computations we study the classical hard-core monomer-dimer models on m x n plane lattice strips with free boundaries. For an arbitrary number v of monomers (or vacancies), we found a logarithmic correction term in the finite-size correction of the free energy per lattice site. The coefficient of the logarithmic correction term depends on the number of monomers present (v) and the parity of the width n of the lattice strip: the coefficient equals to v when n is odd, and v/2 when n is even. The results are generalizations of the previous results for a single monomer in an otherwise fully packed lattice of dimers. We also study the finite-size correction in the low dimer density limit, where the number of dimers d is fixed. In this case the coefficient of the logarithmic correction term equals to d, for both odd and even n.

Packing dimers on(2p+1)×(2q+1)lattices

We use computational method to investigate the number of ways to pack dimers on m x n lattices, where both m and n are odd. In this case, there is always a single vacancy in the lattices. We show that the dimer configuration numbers on (2k+1) x (2k+1) odd square lattices have some remarkable number-theoretical properties in parallel to those of close-packed dimers on 2k x 2k even square lattices, for which an exact solution exists. Furthermore, we demonstrate that there is an unambiguous logarithmic term in the finite size correction of free energy of odd-by-odd lattice strips with any width n > or =1. This logarithmic term determines the distinct behavior of the free energy of odd square lattices. These findings reveal a deep and previously unexplored connection between statistical physics models and number theory and indicate the possibility that the monomer-dimer problem might be solvable.

Transfer matrices for the partition function of the Potts model on lattice strips with toroidal and Klein-bottle boundary conditions

We present a method for calculating transfer matrices for the q-state Potts model partition functions Z ( G , q , v ) , for arbitrary q and temperature variable v , on strip graphs G of the square (sq), triangular (tri), and honeycomb (hc) lattices of width L y vertices and of arbitrarily great length L x vertices, subject to toroidal and Klein-bottle boundary conditions. For the toroidal case we express the partition function as Z ( Λ , L y × L x , q , v ) = ∑ d = 0 L y ∑ j b j ( d ) ( λ Z , Λ , L y , d , j ) m , where Λ denotes lattice type, b j ( d ) are specified polynomials of degree d in q, λ Z , Λ , L y , d , j are eigenvalues of the transfer matrix T Z , Λ , L y , d in the degree- d subspace, and m = L x ( L x / 2 ) for Λ = sq , tri ( hc ) , respectively. An analogous formula is given for Klein-bottle strips. We exhibit a method for calculating T Z , Λ , L y , d for arbitrary L y . In particular, we find some very simple formulas for the determinant det ( T Z , Λ , L y , d ) , and trace Tr ( T Z , Λ , L y ) . Illustrative examples of our general results are given, including new calculations of transfer matrices for Potts model partition functions on strips of the square, triangular, and honeycomb lattices with toroidal or Klein-bottle boundary conditions.